Optimization provides a valuable framework for thinking about, formulating, and solving many problems in machine learning. Since specialized techniques for the quadratic programming problem arising in support vector classification were developed in the 1990s, there has been more and more cross-fertilization between optimization and machine learning, with the large size and computational demands of machine learning applications driving much recent algorithmic research in optimization. This tutorial reviews the major computational paradigms in machine learning that are amenable to optimization algorithms, then discusses the algorithmic tools that are being brought to bear on such applications. We focus particularly on such algorithmic tools of recent interest as stochastic and incremental gradient methods, online optimization, augmented Lagrangian methods, and the various tools that have been applied recently in sparse and regularized optimization